Question 1: Find the perimeter and area of a rectangle whose length and breadth are and respectively.

Answer:

Dimensions of the rectangle: Length Breadth

Therefore Perimeter

Area

Question 2: A rectangular room floor is in area. If its length is , find its perimeter.

Answer:

Dimensions of the rectangle: Length Let Breadth

Area of rectangle

Hence Breadth

Question 3: Find the length of a diagonal of a rectangle whose adjacent sides are and long.

Answer:

Dimensions of the rectangle: Length Let Breadth

Diagonal of a rectangle

Question 4: Find the length of a diagonal of a square of side .

Answer:

Dimension of a square: Side

Diagonal of a square

Question 5: Find the perimeter of a square the sum of the lengths of whose diagonal is .

Answer:

Dimension of a square: Side

Given: Diagonal of the square

We know diagonal of a square

Perimeter of a square

Question 6: The length and breadth of a room are in the ratio . Its area is . Find its perimeter.

Answer:

Dimensions of the rectangle: Let Length Let Breadth

Given: Area is

Therefore

Hence Length Let Breadth

Therefore Perimeter

Question 7: The diagonal of a square is . Find the diagonal of a square whose area is twice the area of .

Answer:

Given: Diagonal of a square is

If the side of the square

… … … … … (i)

Let the side of the second square

Given:

Diagonal … … … … … (ii)

Substituting (i) in (ii) we get

Diagonal

Question 8: The perimeter of a square is . Find its diagonal.

Answer:

Perimeter

Hence Diagonal

Question 9: Find the area of a square that can be inscribed in a circle of radius .

Answer:

Radius

Hence the length of the side

Therefore Area

Question 10: Find the perimeter of a square the sum of the lengths of whose diagonals is .

Answer:

Let the side be

Given:

Therefore Perimeter

Question 11: The diagonal of a square is . Find its area.

Answer:

Let the side of the square

Given:

Therefore Area

Question 12: Find the area and perimeter of a square plot of land the length of whose diagonal is .

Answer:

Given: Diagonal

Let the side of the square

Therefore

Therefore Area

Hence the Perimeter

Question 13: Find the ratio of the area of a square to that of the square drawn on its diagonal.

Answer:

Let the side of square

Therefore diagonal

Hence the ratio

Question 14: The diagonal of square is . Find the diagonal of square whose area is half of the area of .

Answer:

Let the side of square

Therefore

Let the side of square

Therefore

Diagonal of square

Question 15: The perimeter of a square is . The area of a rectangle is less than the area of the given square. If the length of the rectangle is , find its breadth.

Answer:

Let the side of the square

Therefore

Let the breadth of the rectangle

Therefore

Question 16: The perimeter of one square is and that of another is . Find the perimeter and the diagonal of a square whose area is equal to the sum of the areas of these two squares.

Answer:

Let the side of square 1

Therefore 4

Hence the area of square 1

Let the side of square 2

Therefore 4

Hence area of square 2

Therefore area of square 3

Hence the side of square 3

Therefore the perimeter of square 3

Diagonal of square 3

Question 17: The perimeter of a rectangular card board is . If its breadth is , find the length and area of the card board.

Answer:

Dimensions of the rectangle: Length Breadth

Therefore

Hence area

Question 18: If the sides of two squares are in the ratio , prove that their areas are in the ratio .

Answer:

Side of square 1

Therefore Area of square 1

Side of square 2

Therefore Area of square 2

Therefore ratio of areas

Question 19: In exchange for a square plot one of whose sides is , a man wants to buy a rectangular plot long and of the same area as of the square plot. Find the width of the rectangular plot.

Answer:

Side of square plot

Length of rectangular plot

Let breadth of rectangular plot

Therefore

Question 20: A rectangular lawn has two roads each with wide running in the middle of it, one parallel to the length and other parallel to the breadth. Find the cost of graveling them at paisa per square meter.

Answer:

Area of graveled road

Therefore cost of graveling

Question 21: The area of a square plot is hectare. Find the diagonal of the square.

Answer:

Let the side of square plot

Therefore hectare

We know 1 hectare

Therefore

Hence diagonal

Question 22: A lawn is in the form of a rectangle having its sides in the ratio . The area of the lawn is . Find the cost of fencing it at the rate of per meter.

Answer:

Let length and breadth

Therefore

Therefore length and breadth

Therefore perimeter

Therefore cost of fencing

Question 23: The area of a square park is . Find the cost of fencing it at the rate of per meter.

Answer:

Area of square

Let side of the square

Therefore Perimeter

Therefore cost of fencing

Question 24: The area of the base of a rectangular tank is and its sides are in the ratio . Find the cost of planting flowers round it at the rate of per meter.

Answer:

Let length and breadth

Therefore

Therefore length and breadth

Therefore perimeter

Therefore cost of planting flowers

Question 25: A rectangular field meter long has got an area of , what will be the cost of fencing that field on all the four sides, if meter of fencing costs paisa?

Answer:

Length

Area

Therefore breadth

Perimeter

Therefore cost of fencing

Question 26: A rectangular grassy plot is . It has gravel path wide all around it on the inside. Find the area of the path and the cost of constructing it at the rate of per sq. meter.

Answer:

Dimensions of park: Length , Breadth

Inner Dimensions of park: Length , Breadth

Area of path

Therefore cost of construction

Question 27: There is a square field whose side is . A flowerbed is prepared in its center, leaving a gravel path of uniform width all around the flower bed. The total cost of laying the flower bed and graveling the path at and per square meter respectively is . Find the width of the gravel path.

Answer:

Let the side of the garden

Therefore the area of the garden

Area of the path

Therefore

Hence the path

Question 28: How many tiles each will be required to pave the footpath wide carried round the outside of a grassy plot by ?

Answer:

Are of footpath

No of times required

Question 29: A room is long, broad and high. It has two doors and windows each . How much will it cost to whitewash the walls of the room at the rate of per square meter?

Answer:

Dimensions of the Room: Length (l) , Breadth and Height

Dimensions of the Doors: Breadth and Height

Dimensions of the Window: Breadth and Height

Area of walls = Lateral Area

Area of Doors and Windows

Area to be painted

Cost of painting

Question 30: A carpet is laid on the floor of a room . There is a border of constant width around the carpet. If the area of the border is , find its width.

Answer:

Dimensions of the Room: Length (l) , Breadth

Area of Border

Therefore

or (this is not possible as cannot be negative)

Question 31: A rectangular courtyard, long and broad, is to be paved exactly -with square tiles, all of the same size. Find the largest size of such a tile and the number of tiles required to pave it.

Answer:

Dimension of courtyard: Length

Breadth

The largest common divisor of both the numbers is

Therefore the dimension of the largest square tile is

Hence the number of tiles required

Question 32: The cost of fencing a square field at paisa per meter is . Find the cost of reaping the field, it the rate of \ paisa per .

Answer:

Perimeter

Therefore side

Therefore Area

Therefore cost of reaping the field

Question 33: A room long and broad is carpeted with a carpet, leaving an uncovered margin all around the room. If the breadth of the carpet is , find its cost at per meter.

Answer:

Dimensions of the Room: Length (l) , Breadth

Dimensions of the Carpet: Length (l) , Breadth

Area

Let the length of the carpet needed

Therefore

Therefore cost

Question 34: The cost of carpeting a room at Per square meter is . The cost of whitewashing the walls at paisa per sq. meter is . The room is wide. Find its height.

Answer:

Cost of paining area of walls

Question 35: A room long and wide is surrounded by a verandah. Find the width of the verandah if it occupies square meters.

Answer:

Let be the width of the varanda

Therefore

or (not possible)

Therefore the width of the verandah

Question 36: Square carpet is spread in the center of a room square leaving a small margin of equal width all around. The total cost of carpeting at and decorating the margin at is . Find the width of the margin.

Answer:

Dimension of square room

Let the width of the margin

Therefore area of carpet

Area of margin

(not possible)

Hence the width of the margin

Question 37: If the length and breadth of a room are increased by the area is increased By . If the length is increased by and breadth is decreased by , the area is decreased by . Find the perimeter of the room.

Answer:

… … … … … i)

… … … … … ii)

From i)

… … … … … iii)

Therefore Perimeter

from ii)

… … … … … iv)

Solving (iii) and (iv) we get

and .

Question 38: A rectangle has twice the area of a square. The length of the rectangle is greater and the width is greater than the side of the square. Find the perimeter of the square.

Answer:

Let the dimension of rectangle be and . And that of the square be .

Therefore and

Therefore Perimeter of square

Question 39: If the perimeter of a rectangular plot is and the length of its diagonal is , find its area.

Answer:

Let the dimension of rectangle be and .

Therefore … … … … … i)

… … … … … ii)

Therefore

Substituting in (ii)

Hence the area is

Question 40: The length of a rectangular garden is more than its breadth. The numerical value of its area is equal to times the numerical value of its perimeter. Find the dimensions of its garden.

Answer:

Area

Perimeter

Therefore

Therefore or (not possible)

Hence . Therefore dimensions are and

Question 41: A wire when bent in the form of an equilateral triangle encloses an area of Find the area enclosed by the same wire when bent to form: (i) a square (ii) a rectangle whose length is more than its width.

Answer:

Therefore Perimeter

i) When bent in a square

Side

Therefore area

ii) Perimeter of rectangle

If Breadth

Then

Therefore dimensions are and

Hence Area